GENERAL FIBONACCI SEQUENCES IN FINITE GROUPS
Hiiseyin Aydin
Atatiirk Universitesi, Fen-Edb. Fakultesi, Matematik Bolumii, 25240-Erzurum/Turkey
Ramazan Dikici
Atatiirk Universitesi, Kazim Karabekir Egt. Fakultesi, Matematik Bolumii, 25240-Erzurum/Turkey
(Submitted August 1996-Final Revision June 1997)
1. INTRODUCTION
The study of Fibonacci sequences in groups began with the earlier work of Wall [7], where
the ordinary Fibonacci sequences in cyclic groups were investigated. Another early contributor to
this field was Vinson , who was particularly interested in ranks of apparition in ordinary Fibonacci
sequences [6]. In the mid eighties, Wilcox extended the problem to abelian groups [8]. Campbell,
Doostie, and Robertson expanded the theory to some finite simple groups [2]. One of the latest
works in this area is [1], where it is shown that the lengths of ordinary 2-step Fibonacci sequences
are equal to the length of the 2-step Fibonacci recurrences in finite nilpotent groups of nilpotency
class 4 and exponent a prime number/?. The theory has been generalized in [3] to the ordinary
3-step Fibonacci sequences in finite nilpotent groups of nilpotency class 2 and exponent/?.
Definition LI: Let H<G, K< G, and K<H. If HIK is contained in the center of GIK, then
H IK is called a central factor of G. A group G is called nilpotent if it has finite series of normal
subgroups G = G0 > Gx > • • • > Gr = 1 such that Gt_x IG, is a central factor of G for each / = 1, 2,
..., r. The smallest possible r is call the nilpotency class of G.
Further details about nilpotent groups and related topics can be found in [4].
Let G be a free nilpotent group of nilpotency class 2 and exponent p. G has a presentation
G = (x, y9 z: xp = 1, yp - 1, zp = 1, z - (y, x) = y~lx~lyx). Suppose that we have integers n and m
and a recurrence relation in this group given by
We assume that p does not divide n. Then we get a definition of a 2-step general standard Fibonacci sequence which will be (0,1, m,n + m2,...) in Z/pZ. Ifp were permitted to divide n, then
the sequence ultimately would be periodic, but would never return to the consecutive pair 0,1.
The length of the standard sequence is k, which we call the Wall number of the sequence, sometimes called the fundamental period of that sequence.
Each element in the group G can be represented uniquely as xaybzc, where a, b, c G Z /pZ.
The group relations give us a law of composition of standard forms
xaybzc-xa'yb'zc' = xa"yb"zc\
where a", b", and c" are given by the following explicit formulas.
We have a" = a + a', b" = b+b', and c" = c + c' + a'b. These product laws are discussed in
more detail in [1]. In order to study this recurrence, we need a closed formula to describe how to
take the next term of the sequence. Let {xay\b zc)n and (xa'yb'zc')m be two elements in G. The
relevant formulas are
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{xaybzc)n{xa'yb'zc')m = xa"yb"zc",
where
a" = na + ma',
and
c" = nc+mcf +mnafb+(n~^n
2
ah+t™'1)™ a'b'.
2
2. THE MAIN RESULT AND PROOF
Let us use vector notation to calculate the sequence. We put (l,0,G) = ($_l,r0,t0) which
corresponds to x, and (0,1,0) = (% r1? tx) which corresponds to y. We demonstrate more vectors
using the above product formula for c" as
(n, m, 0)•= (^i, r2, t2) and mn, m2 + n, mn2 + yl \mn \ = (s2, r3, t3).
We obtain two sequences (/;) and (tt) via our recurrence. Notice that we have sf - nrf for each
integer /. By induction onj, the 7th term of the third component of our sequence of vectors is
h = mnl X 7>-/-ir/2 + (2 f Z ry-/-iW-i +1 J r S ryw-iW+i •
Let us denote the period of the general Fibonacci sequence in the group G by k(G).
Theorem 2.1: Let p > 3 be a prime number. Then, if G is a nontrivial finite p-group of exponent
p and nilpotency class 2, &(G) = k. There are four assumptions that we will insert:
a)
?2 # 0 (mod/?),
6) m + n-1^0
(modp),
2
3
c) n ~m -n-3mn#0
(mod/?),
2
rfj 3w(m + n) # 0 (mod p).
Proof: Let
&-1
/ \
k-i
/
\ &-i
'* = ™l2Zr*-/-i'/2 + 2 wZ/*-/-iW-i + ^ pS'iw-iW+i.
/=o
V/
1=0
^ ' 1=0
where m,neZ/pZ, /? > 2. In order to show &(G) = k, we must check that ^ = ^ + 1 = 0. The
range of all the following sums is the same as above. Since ri+l = mrt + wj^j, we can recast the last
sum to obtain
We separate this sum to the two parts,
O^mJ+fymn^r^
and 02 = ^ ] » + Q W 2 j X ^ - « - , -
We can pull out factors without difficulty. We put
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GENERAL FIBONACCI SEQUENCES IN FINITE GROUPS
ll = mn2+ 2 \mn
and l2 = L p + L p 2
and then set
^i = 11^-1-1^
and
^2 = Z l i-/-iW-i-
Now we have 0X = Z ^ and #2 = / 2 ^ 2 , and we are in a position to show that </>x = 0 and <f>2 = 0.
First, we prove that
Now let us show that
'-.=<-»'*#'
2
If a and f5 are the roots of x -mx-n = 0, then aP = -n and a+P = m. We have, from the
Binet formula,
a7-/?7
or'-p-'
A
r
i=
We multiply r_7 by (a/?)7 to see that
p
a n d
r
-i=
a-p
a -
a-fi
r_, = (-i)'+1^J/;,
(l)
and also we have
1+rt-i = 1 2 -(-")'- 1 -
(2)
7
This formula was known to Somer [5]. By using r_(z+1) = (-1) (^)
/+1
ri+l and (2), we obtain
Z^+iyK-i=Z(-^]TV+^Z'!We will prove that S/;= 0. Since our recurrence relation is rt =rnri_l+nri_2, we deduce that
ET; = wE^_j + «I/;_ 2 . Replace / - 1 by i in the first sum and / - 2 by /" in the second sum on the
right side to yield'
(!» + /!-1)5>,=0.
(3)
Thus, Z^; = 0 unless m +w-1 is congruent to 0 modulo/?. The next step is to show that
so we will be half way through the proof. From the recurrence relation,
We expand this equation to obtain
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Replacing / - 1 by i in the first, second, and third sums, and / - 2 by i in the last sum on the right
side, we obtain
(4)
Now we have
[" - v - ! ] z ( - ^ p'3+3w"Z (-iy + (£fVxK+«!-i)=o.
Using mrt + wj.i = /}+1 and ^•+1/;_1 = /j 2 - (-n)1'1 = J-2 + (-1)'(w)1"1, we obtain
(»-^-i)Z(-iy(^p-3»Z(-iy(0>-3»z^=a
The last sum is zero by (3). Then we have
^-^-i-3")z(-iy^TV=o..
(5)
We multiply (5) by n to see that
1
in - m3 - n- 3^)X("0f-TV = 0.
Finally, we have
Z(-D'(0 + V = O,
(6)
unless n2-m3-n-3mn
is congruent to 0 modulo p. We deduce that $2 = ®- Hence, we have
completed the first part of the proof. Now we prove that the other part of tk is 0. By (1), write
<*.=Z(-iy(^J+1WBy (4), we have
(„2 _nP_ w ) £ (-iy (I)[
+
V+3wV]£ ( - i ^ i j[+V/-i
+3/ W » 3 X(-iy +1 ^) +2 ^i=o.
From (6), we have our first linear equation:
a ^ S C - 1 ^ } VM+3«« 2 ZH)'(^J V i = o-
(7)
Therefore, from the recurrence relation nrt = ri+2 -mri+l and (6), we get
v+i
i
/ 1 v+i
LH)(0*V4Z<-»^)\/
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+
2-^+l)
=0.
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GENERAL FIBONACCI SEQUENCES IN FINITE GROUPS
We exploit this equation to obtain
r
2
i+2ri+l
Replace i + 2 by / in the first, second, and third sums and / +1 by / in the last sum on the left side
to see that
-£l<-ir^)V=o.
The first and last sums vanish by (6). We multiply the equation by n to obtain a second linear
equation
-awSC-iy^J V*-i+3^S(-i)(^) +1 ^-i = o.
(8)
Hence, from the linear equations (7) and (8),
X(-iy(£j+V>j-i=o
(9)
I ( - i ) ' ( ^ ) V i = o,
(io)
and
unless 3mn(m2 +ri)is congruent to 0 modulo p. Replacing i -1 by / in (10),
3w(iif2+ii)X(-l)'^J\i^ = 0.
So we havefinishedthe second part of the proof. Therefore, we have tk - 0.
Similarly,
tM = mn2Y^r^r? + (" jn^r^r^^
+ ( 7 )w2>*wr,#;+1.
From (1), we have
4+1 = ™ 2 £ ( - l ) ' + ( ^
This is the same as
4J.I
= mn2
&-.r(i]^@|'(-.r'(i)'A, + @»| 1 (-r(ijA.
+ ™„'(-l)*«[iJ^+Q«(-l)»«(lJriV1+Qn<-l)**(;J'i,'i«.
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The last three terms are zero by the fact that rk = 0 because the period of the sequence ri is k.
The first three sums are zero by exactly the same argument as in the proof of tk = 0. Hence,
tk+l = 0. To be more explicit, the same restrictions are still valid for tk+l = 0. Thus, the proof of
Theorem 2.1 is completed.
This result has an obvious interpretation in terms of quotients of groups with presentations
similar to those of Fibonacci groups, which is
F(2, r, m,ri)= (xl,x2,...,xr:
x^x^1
= 1, x2%%~ * = I • • •, ^ V ^ f 1 = 1, x»x?jql = 1>.
ACKNOWLEDGMENT
The authors would like to express their appreciation to the anonymous referee for improving
the statement and proof of Theorem 2.1 and for making useful suggestions regarding the presentation of this paper.
MEFERENCES
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Groups." Applications of Fibonacci Numbers 3:27-35. Ed. G. E. Bergum et al. Dordrecht:
Kluwer, 1990.
3. R. Dikici & G. C. Smith. "Recurrences in Finite Groups." Turkish J. Math 19 (1995):32129.
4. P. Hall. "The Edmonton Notes on Nilpotent Groups." Queen Mary College Mathematics
Notes (1979).
5. L. Somer. "The Divisibility Properties of Primary Lucas Recurrences with Respect to
Primes." The Fibonacci Quarterly 18.4 (1980):316-34.
6. J. Vinson. "The Relations of the Period Modulo m to the Rank of Apparition of m in the
Fibonacci Sequence." The Fibonacci Quarterly 1.1 (1963):37-45.
7. D. D. Wall. "Fibonacci Series Modulo m." Amer. Math. Monthly 67 (1960):525-32.
8. H. J. Wilcox. "Fibonacci Sequences of Period n in Groups." The Fibonacci Quarterly 24.4
(1986):356-61.
AMS Classification Number: 11B39
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